Working with metric graphs

David Bolin, Alexandre B. Simas, and Jonas Wallin

2023-05-03

Introduction

Networks such as street or river networks are examples of metric graphs. A compact metric graph \(\Gamma\) consists of a set of finitely many vertices \(\mathcal{V}=\{v_i\}\) and a finite set \(\mathcal{E}=\{e_j\}\) of edges connecting the vertices. Each edge \(e\) is a curve of finite length \(l_e\) that connects two vertices. These curves are parameterized by arc length and a location \(s\in \Gamma\) is a position on an edge, and can thus be represented as a touple \((e,t)\) where \(t\in[0,l_e]\). Compared to regular graphs, where one typically defines functions on the vertices, we are for metric graphs interested in function that are defined on both the vertices and the edges.

In this vignette we will introduce the metric_graph class of the MetricGraph package. This class provides a user friendly representation of metric graphs, and we will show how to use the class to construct and visualize metric graphs, add data to them, and work with functions defined on the graphs.

For details about Gaussian processes and inference on metric graphs, we refer to the Vignettes

Constructing metric graphs

Basic constructions and properties

A metric graph can be constructed in two ways. The first is to specify all edges in the graph as a SpatialLines object using the sp package. To illustrate this, we first load the sp package

library(sp)

and construct the following lines

line1 <- Line(rbind(c(0,0),c(1,0)))
line2 <- Line(rbind(c(0,0),c(0,1)))
line3 <- Line(rbind(c(0,1),c(-1,1)))
theta <- seq(from=pi,to=3*pi/2,length.out = 50)
line4 <- Line(cbind(sin(theta),1+ cos(theta)))
Lines = sp::SpatialLines(list(Lines(list(line1),ID="1"),
                              Lines(list(line2),ID="2"),
                              Lines(list(line3),ID="3"),
                              Lines(list(line4),ID="4")))

We can now create the graph based on the Lines object as follows

graph <- metric_graph$new(lines = Lines)
graph$plot()

The plot function that was used to create the plot above has various parameters to set the sizes and colors of the vertices and edges, and it has a plotly argument to visualize the graph in 3D. For this to work, the plotly library must be installed.

graph$plot(plotly = TRUE, vertex_size = 5, vertex_color = "blue",
           edge_color = "red", edge_width = 2)

It is also important to know that the 2d version of the plot() method returns a ggplot2 object and can be modified as such. For instance:

p <- graph$plot()
p + ggplot2::labs(title = "Metric graph",
          x = "long", y = "lat")

Similarly, the 3d version of the plot() method returns a plotly object that can also be modified. For instance:

p <- graph$plot(plotly = TRUE)
p <- plotly::layout(p, title = "Metric graph", 
              scene = list(xaxis=
              list(title = "Long"),yaxis=list(title = "Lat")))
p

We can now obtain various properties of the graph: The vertex matrix, which specifies the Euclidian coordinates of the vertices is

graph$V
##      [,1] [,2]
## [1,]    0    0
## [2,]    1    0
## [3,]    0    1
## [4,]   -1    1

and the edge matrix that specified the edges of the graph (i.e., which vertices that are connected by lines) is

graph$E
##      [,1] [,2]
## [1,]    1    2
## [2,]    1    3
## [3,]    3    4
## [4,]    1    4

To obtain the geodesic (shortest path) distance between the vertices, we can use the function compute_geodist:

graph$compute_geodist()
graph$geo_dist
## $`__vertices`
##          [,1]     [,2] [,3]     [,4]
## [1,] 0.000000 1.000000    1 1.570729
## [2,] 1.000000 0.000000    2 2.570729
## [3,] 1.000000 2.000000    0 1.000000
## [4,] 1.570729 2.570729    1 0.000000

The second option it to construct the graph using two matrices V and E that specify the locations (in Euclidean space) of the vertices and the edges. In this case, it is assumed that the graph only has straight lines:

V <- rbind(c(0, 0), c(1, 0), c(0, 1), c(-1, 1))
E <- rbind(c(1, 2), c(1, 3), c(3, 4), c(4, 1))
graph2 <- metric_graph$new(V = V, E = E)
graph2$plot()

Tolerances for the merging vertices and lines

The constructor of the graph has one argument tolerance which is used for connecting lines that are close in Euclidean space. Specifically, the tolerance argument is given as a list with three elements:

These options are often needed when constructing graphs based on real data, for example from OpenStreetMap as we will see later. To illustrate these options, suppose that we want to construct a graph from the following three lines:

line1 <- Line(rbind(c(0,0),c(1,0)))
line2 <- Line(rbind(c(0,0.03),c(0,1)))
line3 <- Line(rbind(c(-1,1),c(0.5,0.03)))
lines = SpatialLines(list(Lines(list(line1),ID="1"),
                          Lines(list(line2),ID="2"),
                          Lines(list(line3),ID="3")))
graph3 <- metric_graph$new(lines = lines)
## The graph is disconnected. You can use the function 'graph_components' to obtain the different connected components.
graph3$plot(degree = TRUE)
print(graph3$nV)
## [1] 6

We added the option degree=TRUE to the plot here to visualize the degrees of each vertex. As expected, one sees that all vertices have degree 1, and none of the three lines are connected. If these are streets in a street network, one might suspect that the two vertices at \((0,0)\) and \((0,0.03)\) really should be the same vertex so that the two lines are connected. This can be adjusted by increasing the vertex_vertex tolerance:

graph3 <- metric_graph$new(lines = lines, tolerance = list(vertex_vertex = 0.05))
## The graph is disconnected. You can use the function 'graph_components' to obtain the different connected components.
graph3$plot(degree = TRUE)

One might also want to add the vertex at \((0.5, 0.03)\) as a vertex on the first line, so that the two lines there are connected. This can be done by adjusting the vertex_line tolerance:

graph3 <- metric_graph$new(lines = lines, tolerance = list(vertex_vertex = 0.05,
                                                           vertex_line = 0.1))
graph3$plot(degree = TRUE)

We can see that the vertex at \((0.5,0)\) was indeed connected with the edge from \((0,0)\) to \((1,0)\) and that vertex now has degree 3 since it is connected with three edges. One can also note that the lines object that was used to create the graph is modified internally in the metric_graph object so that the connections are visualized correctly in the sense that all edges are actually shown as connected lines. This modification might take some time for large graphs with many lines, because of this, the lines object is by default modified if the graph has less than 100 lines, and otherwise it is not modified. Whether or not it should be modified can be set via the argument adjust_lines:

graph3 <- metric_graph$new(lines = lines, tolerance = list(vertex_vertex = 0.05,
                                                           vertex_line = 0.1),
                           adjust_lines = FALSE)
graph3$plot(degree = TRUE)

If the lines object is not modified, then the graph is correct but the plot might not show the graph as connected even though it is. Finally, to add a vertex at the intersection between line2 and line3 we can adjust the line_line tolerance:

graph3 <- metric_graph$new(lines = lines, tolerance = list(vertex_vertex = 0.2,
                                                           vertex_line = 0.1,
                                                           line_line = 0.001))
graph3$plot(degree = TRUE)

Now, the structure of the metric graph does not change if we add or remove vertices of degree 2. Because of this,one might want to remove vertices of degree 2 since this can reduce computational costs. This can be done by setting the remove_deg2 argument while creating the graph:

graph3 <- metric_graph$new(lines = lines, tolerance = list(vertex_vertex = 0.2,
                                                           vertex_line = 0.1,
                                                           line_line = 0.001),
                           remove_deg2 = TRUE)
graph3$plot(degree = TRUE)

Understanding coordinates on graphs

The locations of the vertices are specified in Euclidean coordinates. However, when specifying a position on the graph, it is not practical to work with Euclidean coordinates since not all locations in Euclidean space are locations on the graph. It is instead better to specify a location on the graph by the touple \((i, t)\), where \(i\) denotes the number of the edge and \(t\) is the location on the edge. The location \(t\) can either be specified as the distance from the start of the edge (and then takes values between 0 and the length of the edge) or as the normalized distance from the start of the edge (and then takes values between 0 and 1). The function coordinates can be used to convert between coordinates in Euclidean space and locations on the graph. For example the location at distance 0.2 from the start of the second edge is:

graph$coordinates(PtE = matrix(c(2, 0.2), 1,2), normalized = FALSE)
##      [,1] [,2]
## [1,]    0  0.2

In this case, since the edge has length 1, the location of the point at normalized distance 0.2 from the start of the edge is the same:

graph$coordinates(PtE = matrix(c(2, 0.2), 1,2), normalized = TRUE)
##      [,1] [,2]
## [1,]    0  0.2

The function can also be used to find the closest location on the graph for a location in Euclidean space:

graph$coordinates(XY = matrix(c(0, 0.2), 1,2))
##      [,1] [,2]
## [1,]    1    0

In this case, the normalized argument decides whether the returned value should be given in normalized distance or not.

Methods for working with real data

To illustrate the useage of metric_graph on some real data, we use the osmdata package to download data from OpenStreetMap. In the following code, we extract highways in the city of Copenhagen:

library(osmdata)
set_overpass_url("https://maps.mail.ru/osm/tools/overpass/api/interpreter")
call <- opq(bbox = c(12.4,55.5,12.65,55.9))
call <- add_osm_feature(call, key = "highway",value=c("motorway", "primary",
                                                      "secondary"))
data <- osmdata_sp(call)

graph5 <- metric_graph$new(SpatialLines(data$osm_lines@lines))
## The graph is disconnected. You can use the function 'graph_components' to obtain the different connected components.
graph5$plot(vertex_size = 0)

There are a few things to note about data like this. The first is that the coordinates are given in Longitude and Latitude. Because of this, the edge lengths are by default given in degrees, which may result in very small numbers:

range(graph5$edge_lengths)
## [1] 5.575841e-06 1.115746e-01

This may cause numerical instabilities when dealing with random fields on the graph, and it also makes it difficult to interpret results (unless one has a good intuition about distances in degrees). To avoid such problems, it is better to set the longlat argument when constructing the graph:

graph5 <- metric_graph$new(SpatialLines(data$osm_lines@lines), longlat = TRUE)
## The graph is disconnected. You can use the function 'graph_components' to obtain the different connected components.

This tells the constructor that the coordinates are given in Longitude and Latitude and that distances should be calculated in km. So if we now look at the edge lengths, they are given in km:

range(graph5$edge_lengths)
## [1]  0.000556146 11.209690354

The second thing to note is that the constructor gave a warning that the graph is not connected. This might not be ideal for modeling and we may want to study the different connected components separately. If this is not a concern, one can set the argument check_connected = FALSE while creating the graph. In this case the check is not done and no warning message is printed. To construct all connected components, we can create a graph_components object:

graphs <- graph_components$new(SpatialLines(data$osm_lines@lines), longlat = TRUE)

The graph_components class contains a list of metric_graph objects, one for each connected component. In this case, we have

graphs$n
## [1] 11

components in total, and their total edge lengths in km at

graphs$lengths
##  [1] 176.925387 113.105067  37.255716  10.699136  10.636774   9.353480   8.606712   8.483330   4.209158   4.122888   1.887954

To plot all of them, we can use the plot command of the class:

graphs$plot(vertex_size = 0)

One reason for having multiple components here might be that we set the tolerance for merging nodes too low. In fact, by looking at the edge lengths we see that we have vertices that are as close as

min(graph5$edge_lengths) * 1000
## [1] 0.556146

meters. Let us increase the tolerances so that vertices that are at most a distance of 20 meters (0.02km) apart should be merged, and that vertices that are withing 20 meters of other lines should be merged with them:

graphs <- graph_components$new(SpatialLines(data$osm_lines@lines), longlat = TRUE,
                               tolerance = list(vertex_vertex = 0.02,
                                                vertex_line = 0.02))
graphs$plot(vertex_size = 0)

With this choice, we see that the graph is connected:

graphs$n
## [1] 1

However, since the adjust_vertices argument by default is FALSE for this size of graphs, the plot of the graph might look disconnected in places where vertices have been merged. We can retrieve the graphas a standard metric_graph object to work with in further analysis via the get_largest command:

graph5 <- graphs$get_largest()

Adding data to the graph

Given that we have constructed the metric graph, we can now add data to it. As an example, let us consider the first graph again and suppose that we have observations at a distance 0.5 from the start of each edge. One way of specifying this is as follows

obs.loc <- cbind(1:4, rep(0.5, 4))
obs <- c(1,2,3,4)
df_graph <- data.frame(y = obs, edge_number = obs.loc[,1], 
                        distance_on_edge = obs.loc[,2])
graph$add_observations(data = df_graph)
graph$plot(data = "y", data_size = 2)

In certain situations, it might be easier to specify the relative distances on the edges, so that 0 represents the start and 1 the end of the edge (instead of the edge length). To do so, we can simply specify normalized = TRUE when adding the observations. For example, let us add one more observation at the midpoint of the fourth edge:

obs.loc <- matrix(c(4, 0.5),1,2)
obs <- c(5)
df_new <- data.frame(y=obs, edge_number = obs.loc[,1], 
                          distance_on_edge = obs.loc[,2])
graph$add_observations(data=df_new, normalized = TRUE)
graph$plot(data = "y")

An alternative method is to specify the observations as spatial points objects, where the locations are given in Euclidean coordinates. In this case the observations are added to the closes location on the graph:

obs.loc <- rbind(c(0.7, 0), c(0, 0.2))
obs <- c(6,7)
points <- SpatialPointsDataFrame(coords = obs.loc,
                                 data = data.frame(y = obs))
graph$add_observations(points)
graph$plot(data = "y")

If we want to replace the data in the object, we can use clear_observations() to remove all current data:

graph$clear_observations()

The metric_graph object can hold multiple variables of data, and one can also specify a group argument that it useful for specifying that data is grouped, which for example is the case with data observed at different time points. As an example, let us add two variables observed at two separate time points:

obs.loc <- cbind(c(1:4, 1:4), c(rep(0.5, 4), rep(0.7, 4)))
df_rep <- data.frame(y = c(1, 2, NA, 3, 4, 6, 5, 7),
                     x = c(NA, 8, 9, 10, 11, 12, 13, NA),
                     edge_number = obs.loc[,1], 
                     distance_on_edge = obs.loc[,2],
                     time = c(rep(1, 4), rep(2, 4)))
graph$add_observations(data = df_rep, group = "time")

If NA is given in some variable, this indicates that this specific variable is not measured at that location and group. We can now plot the individual variables by specifying their names in the plot function together with the group number that we want to see. By default the first group is shown.

graph$plot(data = "y", group = 2)

In some cases, we might want to add the observation locations as vertices in the graph. This can be done as follows:

graph$observation_to_vertex()
graph$plot(data = "x", group = 1)

One can note that the command adds all observation locations, from all groups as vertices.

Working with functions on metric graphs

When working with data on metric graphs, one often wants to display functions on the graph. The best way to visualize functions on the graph is to evaluate them on a fine mesh over the graph and then use plot_function. To illustrate this procedure, let us consider the following graph:

V <- rbind(c(0, 0), c(1, 0), c(1, 1), c(0, 1), c(-1, 1), c(-1, 0), c(0, -1))
E <- rbind(c(1, 2), c(2, 3), c(3, 4), c(4, 5),
           c(5, 6), c(6, 1), c(4, 1),c(1, 7))
graph <- metric_graph$new(V = V, E = E)
graph$build_mesh(h = 0.5)
graph$plot(mesh=TRUE)

In the command build_mesh, the argument h decides the largest spacing between nodes in the mesh. As can be seen in the plot, the mesh is very coarse, so let’s reduce the value of h and rebuild the mesh:

  graph$build_mesh(h = 0.01)

Suppose now that we want to display the function \(f(x, y) = x^2 - y^2\) on this graph. We then first evaluate it on the vertices of the mesh and then use the function plot_function to display it:

x <- graph$mesh$V[, 1]
y <- graph$mesh$V[, 2]
f <- x^2 - y^2
graph$plot_function(f)

Alternatively, we can set plotly = TRUE in the plot command to get a 3D visualization of the function:

graph$plot_function(f, plotly = TRUE)

When the first argument of plot_function is a vector, the function assumes that the values in the vector are the values of the function evaluated at the vertices of the mesh. As an alternative, one can also provide the first argument as a matrix consisting of the triplets \((i, t, f(i, t))\). Here \(i\) denotes the edge number, \(t\) the location on the edge, and \(f(i, t)\) the value at that point. To illustrate this, let us first construct a set of locations that are evenly spaced over each edge, then convert these to Euclidean coordinates so that we can evaluate the function above, and finally plot the result:

n.e <- 30
PtE <- cbind(rep(1:graph$nE, each = n.e), 
             rep(seq(from = 0, to = 1, length.out = n.e), graph$nE))
XY <- graph$coordinates(PtE)
f <- XY[, 1]^2 - XY[, 2]^2
graph$plot_function(cbind(PtE, f), plotly = TRUE)